Euclid has a triangle in mind, Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side?
Explanation:
Let the perpendicular on the longest side from the other vertices be h.
∴12×20×h = 80
∴ h = 8
The perpendicular has two triangles on its two sides. On its left, there is one with a hypotenuse of 10. If the two sides are 10 and 8, the third one must be 6.
∴ The base of the other triangle is 20 − 6 = 14
The two sides being 8 and 14, the hypotenuse must be 142+82=260
Hence, option (a).
Alternatively,
Let the third side of the triangle be x.
S=(20+10+x)2=(30+x)2
Area of triangle = s(s-a)(s-b)(s-c)
Area of triangle = 80 = 30+x2×x-102×x+102×30-x2
∴802=(302-x2)×(x2-100)4×4
(900-x2)(x2-100) = 6400 × 16 = 102400
Using options, we get, x = 260
(900 − 260) × 160 = 102400
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